Hologram calculation for compact head-up display

A method of calculating a sub-hologram of a virtual image point for an optical system comprising a display device arranged to display the sub-hologram and a waveguide arranged to replicate the sub-hologram, wherein the method comprises:

• propagating a light wave from the respective virtual image point towards a viewer;
• defining an area of intersection of the propagating light wave at a virtual surface between the virtual image and the viewer, wherein the virtual surface comprises the display device and at least a first virtual replica of the display and a second virtual replica of the display device, each of the virtual replicas being formed by the waveguide, wherein the defined area is bounded by straight line paths from the respective virtual image point to the perimeter of the entrance pupil of the viewer's eye;
• determining at least a first area component of the defined area of the light wave on the first virtual replica of the display device and a second area component of the defined area of the light wave on the second virtual replica of the display device;
• wherein each of the area components of the light wave forms a respective component of the sub-hologram;
• and
• applying a local phase-ramp function to at least one of the first area component and second area component.